This chapter examines equations of motion that are alternatives to the standard Maxwell-Lorentz theory. It argues that there is no consistent and conceptually unproblematic theory that covers ...
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This chapter examines equations of motion that are alternatives to the standard Maxwell-Lorentz theory. It argues that there is no consistent and conceptually unproblematic theory that covers particle-field phenomena. The only consistent and well-behaved electromagnetic theory is not a particle theory at all — the theory of charged dusts; this theory is incompatible with the existence of charged particles on its own. The kind of equations that physicists proposed as ‘fundamental’ or ‘exact’ equations, and their reasons for supporting these equations indicate that there is no sharp distinction between a foundational project aimed at finding coherent and true representations of what is physically possible, and a project concerned with practical, useful representations of particular phenomena.Less

In Search of Coherence

Mathias Frisch

Published in print: 2005-05-26

This chapter examines equations of motion that are alternatives to the standard Maxwell-Lorentz theory. It argues that there is no consistent and conceptually unproblematic theory that covers particle-field phenomena. The only consistent and well-behaved electromagnetic theory is not a particle theory at all — the theory of charged dusts; this theory is incompatible with the existence of charged particles on its own. The kind of equations that physicists proposed as ‘fundamental’ or ‘exact’ equations, and their reasons for supporting these equations indicate that there is no sharp distinction between a foundational project aimed at finding coherent and true representations of what is physically possible, and a project concerned with practical, useful representations of particular phenomena.

In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. ...
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In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. The central concepts are expressions for the fluctuating-force kernels as mode-coupling polynomials of the density-fluctuation-correlation functions. The arrested parts of the latter are solutions of a fixed-point equation, which exhibits bifurcation singularities. The simplest ones are generic and degenerate fold bifurcations, which describe liquid–glass transitions. Schematic models are introduced in order to exemplify by elementary calculations different scenarios for the correlation arrest. The transitions in hard-sphere systems and in square-well systems are explained. Evolutions of the dynamics due to the approach of control parameters towards the critical values for the bifurcation points are analysed in order to show that the theoretical results are similar to those observed for the glassy dynamics of liquids.Less

Foundations of the Mode-Coupling Theory for the Evolution of Glassy Dynamics in Liquids

Wolfgang Götze

Published in print: 2008-12-11

In this chapter, mode-coupling equations of motion for a correlation-function description of the dynamics of simple liquids and colloids are derived and their mathematical properties are analysed. The central concepts are expressions for the fluctuating-force kernels as mode-coupling polynomials of the density-fluctuation-correlation functions. The arrested parts of the latter are solutions of a fixed-point equation, which exhibits bifurcation singularities. The simplest ones are generic and degenerate fold bifurcations, which describe liquid–glass transitions. Schematic models are introduced in order to exemplify by elementary calculations different scenarios for the correlation arrest. The transitions in hard-sphere systems and in square-well systems are explained. Evolutions of the dynamics due to the approach of control parameters towards the critical values for the bifurcation points are analysed in order to show that the theoretical results are similar to those observed for the glassy dynamics of liquids.

This book focuses attention on two aspects of equations of motion in general relativity: the motion of extended bodies(stars) and the motion of small black holes. The objective is to offer a guide to ...
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This book focuses attention on two aspects of equations of motion in general relativity: the motion of extended bodies(stars) and the motion of small black holes. The objective is to offer a guide to prospective researchers into these areas of general relativity and to point out open questions and topics that are ripe for further development. It is over forty years since a text on this subject was published and in that time the research area of equations of motion in general relativity has undergone extraordinary development stimulated by the discovery of the binary neutron star PSR 1913+16 in 1974, which was the first isolated gravitating system found in which general relativity plays a fundamental role in describing theoretically its evolution, and more recently by the advent of kilometre size interferometric gravitational wave detectors which are expected to detect gravitational waves produced by coalescing binary neutron stars. Included in the book are novel topics in general relativistic celestial mechanics: choreographic configurations and the relativistic motion of small black holes.Less

Equations of Motion in General Relativity

Hideki AsadaToshifumi FutamasePeter Hogan

Published in print: 2010-12-01

This book focuses attention on two aspects of equations of motion in general relativity: the motion of extended bodies(stars) and the motion of small black holes. The objective is to offer a guide to prospective researchers into these areas of general relativity and to point out open questions and topics that are ripe for further development. It is over forty years since a text on this subject was published and in that time the research area of equations of motion in general relativity has undergone extraordinary development stimulated by the discovery of the binary neutron star PSR 1913+16 in 1974, which was the first isolated gravitating system found in which general relativity plays a fundamental role in describing theoretically its evolution, and more recently by the advent of kilometre size interferometric gravitational wave detectors which are expected to detect gravitational waves produced by coalescing binary neutron stars. Included in the book are novel topics in general relativistic celestial mechanics: choreographic configurations and the relativistic motion of small black holes.

Heinrich Hertz's strategy for studying the motion of mechanical systems was to deal first with free systems, to which the fundamental law applies, and then with unfree systems considered as ...
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Heinrich Hertz's strategy for studying the motion of mechanical systems was to deal first with free systems, to which the fundamental law applies, and then with unfree systems considered as subsystems of free systems. This chapter discusses Hertz's treatment of free systems, which he dealt with in two steps. First, he discussed the purely geometric properties of straightest paths, then went on to investigate the dynamic theory — that is, how systems move in time. At each step he deduced the general differential equations, derived differential and integral principles, and dealt with the special phenomena of holonomic systems. Hertz's introduction of the concept of force, which in a sense is the highlight of the physical content of his mechanics, is explored by focusing on the differential equations of motion and those general differential principles of mechanics that follow from them.Less

Free systems

JESPER LÜTZEN

Published in print: 2005-05-12

Heinrich Hertz's strategy for studying the motion of mechanical systems was to deal first with free systems, to which the fundamental law applies, and then with unfree systems considered as subsystems of free systems. This chapter discusses Hertz's treatment of free systems, which he dealt with in two steps. First, he discussed the purely geometric properties of straightest paths, then went on to investigate the dynamic theory — that is, how systems move in time. At each step he deduced the general differential equations, derived differential and integral principles, and dealt with the special phenomena of holonomic systems. Hertz's introduction of the concept of force, which in a sense is the highlight of the physical content of his mechanics, is explored by focusing on the differential equations of motion and those general differential principles of mechanics that follow from them.

The Moon — caught in an exact balance of attractive and repulsive forces — orbits the Earth with flawless regularity, never faltering. A ball rolls predictably down a slope of a certain grade, always ...
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The Moon — caught in an exact balance of attractive and repulsive forces — orbits the Earth with flawless regularity, never faltering. A ball rolls predictably down a slope of a certain grade, always passing the same landmarks in the same sequence of times. There are no surprises in the macroscopic world of classical mechanics, a world where to know the present is to predict the future and retrace the past. Observations may differ superficially according to individual frames of reference, but there is always agreement on the larger issues. Energy is conserved. Momentum is conserved. Differences in coordinates and velocities, along with perceptions of motion and rest, are all reconciled, and Newton’s deterministic equations of motion are the law of the land.Less

Three-Part Invention

Michael Munowitz

Published in print: 2006-01-12

The Moon — caught in an exact balance of attractive and repulsive forces — orbits the Earth with flawless regularity, never faltering. A ball rolls predictably down a slope of a certain grade, always passing the same landmarks in the same sequence of times. There are no surprises in the macroscopic world of classical mechanics, a world where to know the present is to predict the future and retrace the past. Observations may differ superficially according to individual frames of reference, but there is always agreement on the larger issues. Energy is conserved. Momentum is conserved. Differences in coordinates and velocities, along with perceptions of motion and rest, are all reconciled, and Newton’s deterministic equations of motion are the law of the land.

A small charged black hole moving in external gravitational and electromagnetic fields is useful for presenting our approach to deriving its equations of motion from the Einstein‐Maxwell field ...
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A small charged black hole moving in external gravitational and electromagnetic fields is useful for presenting our approach to deriving its equations of motion from the Einstein‐Maxwell field equations as it provides a relatively accessible example illustrating how an external 4‐force enters the equations of motion as well as a radiation reaction 4‐force in addition to so‐called tail terms exhibiting retardation effects. As such it is a template for more physically realistic applications of the method, in particular to the equations of motion of a small Schwarzschild black hole moving in an external gravitational field, which is partially included in the current illustrative example when the charge is put equal to zero.Less

Small charged black holes: equations of motion

H. AsadaT. FutamaseP. A. Hogan

Published in print: 2010-12-01

A small charged black hole moving in external gravitational and electromagnetic fields is useful for presenting our approach to deriving its equations of motion from the Einstein‐Maxwell field equations as it provides a relatively accessible example illustrating how an external 4‐force enters the equations of motion as well as a radiation reaction 4‐force in addition to so‐called tail terms exhibiting retardation effects. As such it is a template for more physically realistic applications of the method, in particular to the equations of motion of a small Schwarzschild black hole moving in an external gravitational field, which is partially included in the current illustrative example when the charge is put equal to zero.

This section presents annotations of the manuscript of Albert Einstein's canonical 1916 paper on the general theory of relativity. It begins with a discussion of the foundation of the general theory ...
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This section presents annotations of the manuscript of Albert Einstein's canonical 1916 paper on the general theory of relativity. It begins with a discussion of the foundation of the general theory of relativity, taking into account Einstein's fundamental considerations on the postulate of relativity, and more specifically why he went beyond the special theory of relativity. It then considers the spacetime continuum, explaining the role of coordinates in the new theory of gravitation. It also describes tensors of the second and higher ranks, multiplication of tensors, the equation of the geodetic line, the formation of tensors by differentiation, equations of motion of a material point in the gravitational field, the general form of the field equations of gravitation, and the laws of conservation in the general case. Finally, the behavior of rods and clocks in the static gravitational field is examined.Less

The Annotated Manuscript

Hanoch GutfreundJürgen Renn

Published in print: 2017-05-09

This section presents annotations of the manuscript of Albert Einstein's canonical 1916 paper on the general theory of relativity. It begins with a discussion of the foundation of the general theory of relativity, taking into account Einstein's fundamental considerations on the postulate of relativity, and more specifically why he went beyond the special theory of relativity. It then considers the spacetime continuum, explaining the role of coordinates in the new theory of gravitation. It also describes tensors of the second and higher ranks, multiplication of tensors, the equation of the geodetic line, the formation of tensors by differentiation, equations of motion of a material point in the gravitational field, the general form of the field equations of gravitation, and the laws of conservation in the general case. Finally, the behavior of rods and clocks in the static gravitational field is examined.

This chapter presents an account of magnetization field as a continuous function of position, governed by classical equations of motion. It discusses equations of motion, damping, and approaching the ...
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This chapter presents an account of magnetization field as a continuous function of position, governed by classical equations of motion. It discusses equations of motion, damping, and approaching the Curie temperature.Less

THE CLASSICAL MAGNETIZATION FIELD

HARRY SUHL

Published in print: 2007-06-21

This chapter presents an account of magnetization field as a continuous function of position, governed by classical equations of motion. It discusses equations of motion, damping, and approaching the Curie temperature.

The dynamics of fluids deals with the relation (the equation of motion of the fluid) between the stresses on the surface of fluid elements, the volume forces and the rate of strain introduced in ...
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The dynamics of fluids deals with the relation (the equation of motion of the fluid) between the stresses on the surface of fluid elements, the volume forces and the rate of strain introduced in Chapter 3. For viscous “real” fluids, the stresses correspond to pressure and viscous forces; for so-called Newtonian fluids, the rates of strain and the viscous stresses are proportional with a constant factor called viscosity. The equation of motion is first established in the general case and reduces to the Navier–Stokes equation for Newtonian fluids. The boundary conditions at the walls bounding the flow are then discussed. Finally, this chapter considers the rheology of non-Newtonian fluids, for which the relationship between the stress and the rate of strain is no longer linear and/or instantaneous. These results are then applied to a few examples of parallel flows without convective terms of Newtonian as well as non-Newtonian fluids.Less

Dynamics of viscous fluids: rheology and parallel flows

Etienne GuyonJean-Pierre HulinLuc PetitCatalin.D. Mitescu

Published in print: 2015-01-20

The dynamics of fluids deals with the relation (the equation of motion of the fluid) between the stresses on the surface of fluid elements, the volume forces and the rate of strain introduced in Chapter 3. For viscous “real” fluids, the stresses correspond to pressure and viscous forces; for so-called Newtonian fluids, the rates of strain and the viscous stresses are proportional with a constant factor called viscosity. The equation of motion is first established in the general case and reduces to the Navier–Stokes equation for Newtonian fluids. The boundary conditions at the walls bounding the flow are then discussed. Finally, this chapter considers the rheology of non-Newtonian fluids, for which the relationship between the stress and the rate of strain is no longer linear and/or instantaneous. These results are then applied to a few examples of parallel flows without convective terms of Newtonian as well as non-Newtonian fluids.

A technique for extracting the relativistic equations of motion of Schwarzschild, Reissner–Nordström, or Kerr particles moving in external fields, from the vacuum Einstein or Einstein–Maxwell field ...
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A technique for extracting the relativistic equations of motion of Schwarzschild, Reissner–Nordström, or Kerr particles moving in external fields, from the vacuum Einstein or Einstein–Maxwell field equations, as appropriate, is motivated and explained. The distinguishing features of the technique are (1) the use of a coordinate system attached to the null hypersurface histories of the wavefronts of the radiation produced by the particle motion, (2) the emergence of the approximate equations of motion of the particle from the field equations and the requirement that the wavefronts are smoothly deformed spheres near the particle, and (3) the absence of infinities arising in the process. The particular case of a spinning particle or Kerr particle is described in detail. The particles are not test particles although spinning test particles are also discussed.Less

Equations of motion

Published in print: 2013-05-23

A technique for extracting the relativistic equations of motion of Schwarzschild, Reissner–Nordström, or Kerr particles moving in external fields, from the vacuum Einstein or Einstein–Maxwell field equations, as appropriate, is motivated and explained. The distinguishing features of the technique are (1) the use of a coordinate system attached to the null hypersurface histories of the wavefronts of the radiation produced by the particle motion, (2) the emergence of the approximate equations of motion of the particle from the field equations and the requirement that the wavefronts are smoothly deformed spheres near the particle, and (3) the absence of infinities arising in the process. The particular case of a spinning particle or Kerr particle is described in detail. The particles are not test particles although spinning test particles are also discussed.

This chapter presents two critical points to bear in mind right from the start: (1) never take for granted the apparent willingness of nature to yield to reason, and (2) never expect to find a ...
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This chapter presents two critical points to bear in mind right from the start: (1) never take for granted the apparent willingness of nature to yield to reason, and (2) never expect to find a one-size-fits-all equation of motion or theory of everything under the Sun. Different phenomena call for different models, and the secret of scientific success is to develop a sense of reasonableness and proportion — an ability to appreciate scale and scope, to ask the right questions, to make the appropriate simplifications.Less

Great Expectations

Michael Munowitz

Published in print: 2006-01-12

This chapter presents two critical points to bear in mind right from the start: (1) never take for granted the apparent willingness of nature to yield to reason, and (2) never expect to find a one-size-fits-all equation of motion or theory of everything under the Sun. Different phenomena call for different models, and the secret of scientific success is to develop a sense of reasonableness and proportion — an ability to appreciate scale and scope, to ask the right questions, to make the appropriate simplifications.

The book presents a self-contained exposition of the mode-coupling theory for the evolution of glassy dynamics in liquids. This theory is based on polynomial expressions for the correlations of force ...
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The book presents a self-contained exposition of the mode-coupling theory for the evolution of glassy dynamics in liquids. This theory is based on polynomial expressions for the correlations of force fluctuations in terms of those of density fluctua-tions. These mode-coupling polynomials are motivated as descriptions of the cage-effect-induced transient localization of particles in condensed matter. It is proven that the implied regular mode-coupling equations of motion determine uniquely models for a correlation-function description of the dynamics. This holds for all choices of the polynomial coefficients, which serve as coupling constants. The arrested parts of the correlations are solutions of fixed-point equations. They exhibit spontaneous singularities, which are equivalent to the bifurcation singularities of the real roots of real polynomials. They deal with idealized liquid-glass and glass-glass transitions. Driving the coupling constants towards their critical values, the correlation functions exhibit the evolution of complex dynamics. Its subtleties are due to the interplay of nonlinearities and divergent retardation effects. The book discusses that the relaxation features are similar to those observed in experimental and molecular-dynamics-simulation studies of con-ventional liquids and colloids. Asymptotic expansions are derived for the mode-coupling-theory functions for small frequencies and small separations of the coupling constants from the transition values. The leading-order asymptotic contributions provide an understanding of the essential facets of the scenarios. The leading-asymptotic corrections are deduced and applied to quantify the evolution of the leading-order description.Less

Complex Dynamics of Glass-Forming Liquids : A Mode-Coupling Theory

Wolfgang Götze

Published in print: 2008-12-11

The book presents a self-contained exposition of the mode-coupling theory for the evolution of glassy dynamics in liquids. This theory is based on polynomial expressions for the correlations of force fluctuations in terms of those of density fluctua-tions. These mode-coupling polynomials are motivated as descriptions of the cage-effect-induced transient localization of particles in condensed matter. It is proven that the implied regular mode-coupling equations of motion determine uniquely models for a correlation-function description of the dynamics. This holds for all choices of the polynomial coefficients, which serve as coupling constants. The arrested parts of the correlations are solutions of fixed-point equations. They exhibit spontaneous singularities, which are equivalent to the bifurcation singularities of the real roots of real polynomials. They deal with idealized liquid-glass and glass-glass transitions. Driving the coupling constants towards their critical values, the correlation functions exhibit the evolution of complex dynamics. Its subtleties are due to the interplay of nonlinearities and divergent retardation effects. The book discusses that the relaxation features are similar to those observed in experimental and molecular-dynamics-simulation studies of con-ventional liquids and colloids. Asymptotic expansions are derived for the mode-coupling-theory functions for small frequencies and small separations of the coupling constants from the transition values. The leading-order asymptotic contributions provide an understanding of the essential facets of the scenarios. The leading-asymptotic corrections are deduced and applied to quantify the evolution of the leading-order description.

All throughout nature, there are systems simple in every way that nevertheless behave chaotically and unpredictably: small systems, systems with few interactions, deterministic systems with the most ...
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All throughout nature, there are systems simple in every way that nevertheless behave chaotically and unpredictably: small systems, systems with few interactions, deterministic systems with the most elementary equations of motion. From dripping water faucets to the population of wolves in a forest, from the frustrating unpredictability of the weather to the rings of Saturn, nonlinear feedback makes the simple complex. Determinism carried too far leads to the butterfly effect and chaos.Less

Surprise Endings

Michael Munowitz

Published in print: 2006-01-12

All throughout nature, there are systems simple in every way that nevertheless behave chaotically and unpredictably: small systems, systems with few interactions, deterministic systems with the most elementary equations of motion. From dripping water faucets to the population of wolves in a forest, from the frustrating unpredictability of the weather to the rings of Saturn, nonlinear feedback makes the simple complex. Determinism carried too far leads to the butterfly effect and chaos.

This book is aimed at students who have completed a final year undergraduate course on general relativity and supplemented it with additional techniques by individual study or in a taught MSc ...
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This book is aimed at students who have completed a final year undergraduate course on general relativity and supplemented it with additional techniques by individual study or in a taught MSc programme. The additional technical knowledge required involves the Cartan calculus, the tetrad formalism including aspects of the Newman–Penrose formalism, the Ehlers–Sachs theory of null geodesic congruences, and the Petrov classification of gravitational fields. Each chapter could be used as a basis for an advanced undergraduate or early postgraduate project. The topics covered fall under three general headings: Gravitational waves in vacuo and in a cosmological setting, equations of motion with particular emphasis on spinning particles, and black holes. These are not individual applications of the techniques mentioned above. The techniques are available for use in whole or in part (mainly in part) as each situation demands.Less

Claude BarrabèsPeter A. Hogan

Published in print: 2013-05-23

This book is aimed at students who have completed a final year undergraduate course on general relativity and supplemented it with additional techniques by individual study or in a taught MSc programme. The additional technical knowledge required involves the Cartan calculus, the tetrad formalism including aspects of the Newman–Penrose formalism, the Ehlers–Sachs theory of null geodesic congruences, and the Petrov classification of gravitational fields. Each chapter could be used as a basis for an advanced undergraduate or early postgraduate project. The topics covered fall under three general headings: Gravitational waves in vacuo and in a cosmological setting, equations of motion with particular emphasis on spinning particles, and black holes. These are not individual applications of the techniques mentioned above. The techniques are available for use in whole or in part (mainly in part) as each situation demands.

From the information obtained in DFT, in particular the magnetic moments and the Heisenberg exchange parameters, one has the possibility to make a connection to atomistic spin-dynamics. In this ...
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From the information obtained in DFT, in particular the magnetic moments and the Heisenberg exchange parameters, one has the possibility to make a connection to atomistic spin-dynamics. In this chapter the essential features of this connection is described. It is also discussed under what length and time-scales that this approach is a relevant approximation. The master equation of atomistic spin-dynamics is derived, and discussed in detail. In addition we give examples of how this equation describes the magnetization dynamics of a few model systems.Less

The Atomistic Spin Dynamics Equation of Motion

Olle ErikssonAnders BergmanLars BergqvistJohan Hellsvik

Published in print: 2017-02-23

From the information obtained in DFT, in particular the magnetic moments and the Heisenberg exchange parameters, one has the possibility to make a connection to atomistic spin-dynamics. In this chapter the essential features of this connection is described. It is also discussed under what length and time-scales that this approach is a relevant approximation. The master equation of atomistic spin-dynamics is derived, and discussed in detail. In addition we give examples of how this equation describes the magnetization dynamics of a few model systems.

Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle ...
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Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ‎ and the T-matrixLess

Equations of Motion with Particle–Particle Interactions and Approximations

Norman J. Morgenstern Horing

Published in print: 2017-07-27

Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ‎ and the T-matrix

This chapter discusses the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics. In physics, action is a mathematical functional which ...
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This chapter discusses the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics. In physics, action is a mathematical functional which takes the trajectory, or path, of the system as its argument and has a real number as its result. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more generally, is stationary. The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral. The chapter first provides an overview of Lagrangians, action, and Hamiltonians in order to draw out an alternative approach to finding equations of motion. It then considers the classical wave equation and classical scattering and concludes with an analysis of the classical inverse scattering problem.Less

The Classical Connection

John A. Adam

Published in print: 2017-05-30

This chapter discusses the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics. In physics, action is a mathematical functional which takes the trajectory, or path, of the system as its argument and has a real number as its result. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more generally, is stationary. The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral. The chapter first provides an overview of Lagrangians, action, and Hamiltonians in order to draw out an alternative approach to finding equations of motion. It then considers the classical wave equation and classical scattering and concludes with an analysis of the classical inverse scattering problem.

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and ...
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This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.Less

The post-Newtonian approximation

Nathalie DeruelleJean-Philippe Uzan

Published in print: 2018-08-30

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

This chapter begins by finding the field created by compact objects in the post-linear approximation of general relativity. The second quadrupole formula is then completely proven. Next, the chapter ...
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This chapter begins by finding the field created by compact objects in the post-linear approximation of general relativity. The second quadrupole formula is then completely proven. Next, the chapter finds the equations of motion of the bodies in the field which they create to second order in the perturbations, assuming that their velocities are small. It shows that, to correctly describe the radiation reaction at 2.5 PN order, it will prove necessary to iterate Einstein equations a third time. This leads the discussion to the equations of motion, which generalize to order 1/c5 the EIH equations of order 1/c⁲. Finally, the chapter studies the effect of the radiation reaction force on the sources, and shows that there is an energy balance at 2.5 PN order between the energy radiated to infinity and the mechanical energy lost by the system.Less

The two-body problem and radiative losses

Nathalie DeruelleJean-Philippe Uzan

Published in print: 2018-08-30

This chapter begins by finding the field created by compact objects in the post-linear approximation of general relativity. The second quadrupole formula is then completely proven. Next, the chapter finds the equations of motion of the bodies in the field which they create to second order in the perturbations, assuming that their velocities are small. It shows that, to correctly describe the radiation reaction at 2.5 PN order, it will prove necessary to iterate Einstein equations a third time. This leads the discussion to the equations of motion, which generalize to order 1/c5 the EIH equations of order 1/c⁲. Finally, the chapter studies the effect of the radiation reaction force on the sources, and shows that there is an energy balance at 2.5 PN order between the energy radiated to infinity and the mechanical energy lost by the system.